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Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis is devoted to quantitative questions about the discrete spectrum of Schrödinger-type operators.

In Paper I we show that the Lieb-Thirring inequalities on moments of negative eigen¬values remain true, with possibly different constants, when the critical Hardy weight is subtracted from the Laplace operator.

In Paper II we prove that the one-dimensional analog of this inequality holds even for the critical value of the moment parameter. In Paper III we establish Hardy-Lieb-Thirring inequalities for fractional powers of the Laplace operator and, in particular, relativistic Schrödinger operators. We do so by first establishing Hardy-Sobolev inequalities for such operators. We also allow for the inclu¬sion of magnetic fields.

As an application, in Paper IV we give a proof of stability of relativistic matter with magnetic fields up to the critical value of the nuclear charge.

In Paper V we derive inequalities for moments of the real part and the modulus of the eigen¬values of Schrödinger operators with complex-valued potentials.

Place, publisher, year, edition, pages
Stockholm: KTH , 2007. , vii, 28 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 07:03
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4344ISBN: 978-91-7178-626-5 (print)OAI: oai:DiVA.org:kth-4344DiVA: diva2:11908
Public defence
2007-05-09, Sal F3, KTH, Lindstedtsvägen 26, Stockholm, 09:00
Opponent
Supervisors
Note
QC 20100708Available from: 2007-04-25 Created: 2007-04-25 Last updated: 2010-07-08Bibliographically approved
List of papers
1. On Lieb-Thirring inequalities for Schrödinger operators with virtual level
Open this publication in new window or tab >>On Lieb-Thirring inequalities for Schrödinger operators with virtual level
2006 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 264, no 3, 725-740 p.Article in journal (Refereed) Published
Abstract [en]

We consider the operator H = - Delta- V in L-2(R-d), d >= 3. For the moments of its negative eigenvalues we prove the estimate

tr H--(gamma) <= C-gamma,C-d integral(Rd) (V(x) - (d-2)(2)/4\x\(2))(gamma+d/2) dx, gamma > 0.

Similar estimates hold for the one-dimensional operator with a Dirichlet condition at the origin and for the two-dimensional Aharonov-Bohm operator.

Keyword
BOUND-STATES; NUMBER
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7026 (URN)10.1007/s00220-006-1521-z (DOI)000237193800008 ()2-s2.0-33646531642 (Scopus ID)
Note
QC 20100708Available from: 2007-04-25 Created: 2007-04-25 Last updated: 2017-12-14Bibliographically approved
2. Lieb-Thirring inequalities on the half-line with critical exponent
Open this publication in new window or tab >>Lieb-Thirring inequalities on the half-line with critical exponent
2008 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 10, no 3, 739-755 p.Article in journal (Refereed) Published
Abstract [en]

We consider the operator -d(2)/dr(2) - V in L-2(R+) with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound for any alpha is an element of [0, 1) and gamma >= (1 - alpha)/2. This includes a Lieb-Thirring inequality in the critical endpoint case.

Keyword
SCHRODINGER-OPERATORS
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7027 (URN)000257869200006 ()
Note
QC 20100708. Uppdaterad från Accepted till Published 20100708.Available from: 2007-04-25 Created: 2007-04-25 Last updated: 2017-12-14Bibliographically approved
3. Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators
Open this publication in new window or tab >>Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators
2008 (English)In: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 21, no 4, 925-950 p.Article in journal (Refereed) Published
Keyword
Diamagnetic inequality; Hardy inequality; Lieb-Thirring inequalities; Relativistic Schrödinger operator; Sobolev inequalities; Stability of matter
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7028 (URN)000257550700001 ()2-s2.0-55349101942 (Scopus ID)
Note
QC 20100708Available from: 2007-04-25 Created: 2007-04-25 Last updated: 2017-12-14Bibliographically approved
4. Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value
Open this publication in new window or tab >>Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value
2007 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 275, no 2, 479-489 p.Article in journal (Refereed) Published
Abstract [en]

We give a proof of stability of relativistic matter with magnetic fields all the way up to the critical value of the nuclear charge Z alpha = 2/pi.

Keyword
INSTABILITY; OPERATORS
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7029 (URN)10.1007/s00220-007-0307-2 (DOI)000248911300006 ()2-s2.0-34548238250 (Scopus ID)
Note
QC 20100708. Uppdaterad från Accepted till Published 20100708.Available from: 2007-04-25 Created: 2007-04-25 Last updated: 2017-12-14Bibliographically approved
5. Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials
Open this publication in new window or tab >>Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials
2006 (English)In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 77, no 3, 309-316 p.Article in journal (Refereed) Published
Abstract [en]

Inequalities are derived for power sums of the real part and the modulus of the eigenvalues of a Schrodinger operator with a complex-valued potential.

Keyword
Schrodinger operator; Lieb-Thirring inequalities; complex potential
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7030 (URN)10.1007/s11005-006-0095-1 (DOI)000240126700007 ()2-s2.0-33748086051 (Scopus ID)
Note
QC 20100708Available from: 2007-04-25 Created: 2007-04-25 Last updated: 2017-12-14Bibliographically approved

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